RESEARCH Open Access Local stability of the Pexiderized Cauchy and Jensen’s equations in fuzzy spaces Abbas Najati 1 , Jung Im Kang 2* and Yeol Je Cho 3 * Correspondence: jikang@nims.re. kr 2 National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, 463-1 Jeonmin-dong, Yuseong-gu, Daejeon 305-811, Korea Full list of author information is available at the end of the article Abstract Lex X be a normed space and Y be a Banach fuzzy space. Let D ={(x, y) Î X × X :|| x|| + ||y|| ≥ d} where d > 0. We prove that the Pexiderized Jensen functional equation is stable in the fuzzy norm for functions defined on D and taking values in Y.We consider also the Pexiderized Cauc hy functional equation. 2000 Mathematics Subject Classification: 39B22; 39B82; 46S10. Keywords: Pexiderized Cauchy functional equation, generalized Hyers-Ulam stability, Jensen functional equation, non-Archimedean space 1. Introduction The functional equation (ξ)isstable if any function g satisfying the equation (ξ) approximately is near to the true solution of (ξ). The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms: Let G 1 be a group and let G 2 beametricgroupwiththemetricd(·,·). Given ε >0, does there exist δ > 0 such that if a function h : G 1 ® G 2 satisfies the inequality d(h (xy), h(x)h(y )) <δ for all x, y Î G 1 , then there exists a homomorphism H : G 1 ® G 2 with d(h(x), H(x)) < ε for all x Î G 1 ? In other words, we are looking for situations when the homomorphisms are stable, i. e., if a mapping is almost a homomorphism, then there exists a true homomorphism near it. If we turn our attention to the case of functional equations, then we can ask the question: When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation. In 1941, Hyers [2] gave a partial solution of Ulam’s problem for the ca se of approxi- mate additive mappings under the assumption that G 1 and G 2 are Banach spaces. In 1950, Aoki [3] prov ide d a generalization of the Hyers’ theorem for additive mappings, and in 1978, Th.M. Rassias [4] succeeded in extending the result of H yers for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The stabi- lity phenomenon that was introduced and proved by Th.M. Rassias is called the gener- alized Hyers-Ulam stability. Forti [6] and Gǎvruta [7] have generalized the result of Th.M. Rassias, which permitted the Cauchy difference to become arbitrary unbounded. The stability p roblems of several functional equations have been extensively investi- gated by a number o f authors, and there are many interesting results concerning this problem. A large list of references can be found, for example, in [8-29]. Najati et al. Journal of Inequalities and Applications 2011, 2011:78 http://www.journalofinequalitiesandapplications.com/content/2011/1/78 © 2011 Najati et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Following [30], we give the following notion of a fuzzy norm. Definition 1.1.[30]LetX be a real vector space. A function N : X × ℝ ® [0, 1] is called a fuzzy norm on X if, for all x, y Î X and s, t Î ℝ, (N 1 ) N(x, t) = 0 for all t ≤ 0; (N 2 ) x = 0 if and only if N(x, t) = 1 for all t >0; (N 3 ) N( cx, t)=N(x, t | c | ) if c ≠ 0; (N 4 ) N(x + y, s + t) ≥ min{N(x, s), N(y, t)}; (N 5 ) N(x,·) is a nondecreasing function on ℝ and lim t®∞ N(x, t)=1; (N 6 ) for x ≠ 0, N(x,·) is continuous on ℝ. The pair (X, N) is called a fuzzy normed vector space. Example 1.2. Let (X, ||·||) be a normed linear space and let a, b > 0. Then, N( x , t)= ⎧ ⎨ ⎩ αt αt + βx , t > 0, x ∈ X , 0, t ≤ 0, x ∈ X is a fuzzy norm on X. Example 1.3. Let (X, ||·||) be a normed linear space and let b >a > 0. Then, N( x , t)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0, t ≤ αx, t t +(β − α)x , αx < t ≤ βx ; 1, t >βx is a fuzzy norm on X. Definition 1.4.Let(X , N) be a fuzzy normed space. A sequence {x n }inX is said to be convergent if there exists x Î X such that lim n®∞ N(x n - x, t)=1forallt >0.In this case, x is called the limit of the sequence {x n }, and we denote it by N - lim x n = x. The limit of the convergent sequence {x n }in(X, N) is unique. Since if N - lim x n = x and N-lim x n = y for some x, y Î X, it follows from (N 4 ) that N( x − y, t) ≥ min N x − x n , t 2 , N x n − y, t 2 for all t > 0 and n Î N. So, N(x - y, t) = 1 for all t > 0. Hence, (N 2 ) implies that x = y . Definition 1.5. Let (X, N) be a fuzzy normed space. A sequence {x n }inX is called a Cauchy sequence if, for any ε > 0 and t > 0, there exists M ∈ N such that, for all n ≥ M and p >0, N( x n+ p − x n , t) > 1 − ε . It follows from (N 4 ) that every convergent sequence in a fuzzy normed space is a Cauchy sequence. If, in a fuzzy normed space, every C auchy sequence is convergent, Najati et al. Journal of Inequalities and Applications 2011, 2011:78 http://www.journalofinequalitiesandapplications.com/content/2011/1/78 Page 2 of 8 then the fuzzy norm is said to be complete, and the fuzzy normed space is called a fuzzy Banach space. Example 1.6. [21] Let N : ℝ × ℝ ® [0, 1] be a fuzzy norm on ℝ defined by N( x , t)= ⎧ ⎨ ⎩ t t + |x| , t > 0 , 0, t ≤ 0. Then, (ℝ, N) is a fuzzy Banach space. Recently, several various fuzzy stability results concerning a Cauchy sequence, Jense n and quadratic functional equations were investigated in [17-20]. 2. A loc al Hyers-Ulam stability of Jensen ’s equation In 1998, Jung [16] investigated the Hyers-Ulam stability for Jensen’sequationona restricted domain. In this section, we prove a local Hyers-Ulam stability of the Pexider- ized Jensen functional equation in fuzzy normed spaces. Theorem 2.1. Let X be a normed space,(Y, N) be a fuzzy Banach space, and f, g, h : X® Ybemappingswithf(0) = 0. Suppose that δ >0is a positive real number, and z 0 is a fixed vector of a fuzzy normed space (Z, N’) such that N 2f x + y 2 − g(x) − h(y), t + s ≥ min{N (δz 0 , t), N (δz 0 , s) } (2:1) for all x, y Î X with ||x|| + ||y|| ≥ d and positive real numbers t, s. Then, there exists a unique additive mapping T : X® Y such that N ( f ( x ) − T ( x ) , t ) ≥ N ( 40δz 0 , t ), (2:2) N ( T ( x ) − g ( x ) + g ( 0 ) , t ) ≥ N ( 30δz 0 , t ), (2:3) N ( T ( x ) − h ( x ) + h ( 0 ) , t ) ≥ N ( 30δz 0 , t ) (2:4) for all x Î X and t >0. Proof. Suppose that ||x|| + ||y|| <d holds. If ||x|| + ||y|| = 0, let z Î X with ||z|| = d. Otherwise, z := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (d + x) x x , if x≥y, (d + y) y y , if x < y . It is easy to verify that x − z + y + z≥d, 2z + x − z≥d, y + 2z≥d , y + z + z≥d, x + z≥d. (2:5) Najati et al. Journal of Inequalities and Applications 2011, 2011:78 http://www.journalofinequalitiesandapplications.com/content/2011/1/78 Page 3 of 8 It follows from (N 4 ), (2.1) and (2.5) that N 2f x + y 2 − g(x) − h(y), t + s ≥ min N 2f x + y 2 − g(y + z) − h(x − z), t + s 5 , N 2f x + z 2 − g(2z) − h(x − z), t + s 5 , N 2f y +2z 2 − g(2z) − h(y), t + s 5 , N 2f y +2z 2 − g(y + z) − h(z), t + s 5 , N 2f x + z 2 − g(x) − h(z), t + s 5 ≥ min{N ( 5δz 0 , t ) , N ( 5δz 0 , s ) } for all x, y Î X with ||x|| + ||y|| <d and positive real numbers t, s. Hence, we have N 2f x + y 2 − g(x) − h(y), t + s ≥ min{N (5δz 0 , t), N (5δz 0 , s) } (2:6) for all x, y Î X and positive real numbers t, s. Letting x =0(y = 0) in (2.6), we get N 2f y 2 − g(0) − h(y), t + s ≥ min{N (5δz 0 , t), N (5δz 0 , s)} , N 2f x 2 − g(x) − h(0), t + s ≥ min{N (5δz 0 , t), N (5δz 0 , s)} (2:7) for all x, y Î X and positive real numbers t, s. It follows from (2.6) and (2.7) that N 2f x + y 2 − 2f x 2 − 2f y 2 , t + s ≥ min N 2f x + y 2 − g(x) − h(y), t + s 4 , N 2f x 2 − g(x) − h(0), t + s 4 , N 2f y 2 − g(0) − h(y), t + s 4 , N(g(0) + h(0), t + s 4 ≥ min{N ( 20δz 0 , t ) , N ( 20δz 0 , s ) } for all x, y Î X and positive real numbers t, s. Hence, N f (x + y) − f(x) − f (y), t + s ≥ min{N (10δz 0 , t), N (10δz 0 , s) } (2:8) for all x, y Î X and positive real numbers t, s. Letting y = x an d t = s in (2.8), we infer that N f (2x) 2 − f (x), t ≥ N (10δz 0 , t ) (2:9) for all x Î X and positive real number t. replacing x by 2 n x in (2.9), we get N f (2 n+1 x) 2 n+1 − f (2 n x) 2 n , t 2 n ≥ N (10δz 0 , t ) (2:10) Najati et al. Journal of Inequalities and Applications 2011, 2011:78 http://www.journalofinequalitiesandapplications.com/content/2011/1/78 Page 4 of 8 for all x Î X, n ≥ 0 and positive real number t. It follows from (2.10) that N f (2 n x) 2 n − f (2 m x) 2 m , n− 1 k=m t 2 k ≥ min n− 1 k=m N f (2 k+1 x) 2 k+1 − f (2 k x) 2 k , t 2 k ≥ N ( 10δz 0 , t ) (2:11) for all x Î X, t > 0 and integers n ≥ m ≥ 0. For any s, ε > 0, there exist an integer l > 0 and t 0 > 0 such that N’(10δz 0 , t 0 )>1-ε and n−1 k=m t 0 2 k > s for all n ≥ m ≥ l. Hence, it follows from (2.11) that N f (2 n x) 2 n − f (2 m x) 2 m , s > 1 − ε for all n ≥ m ≥ l.So { f (2 n x) 2 n } is a Cauchy sequence in Y for all x Î X. Since (Y, N)is complete, { f (2 n x) 2 n } converges to a point T(x) Î Y.Thus,wecandefineamappingT : X ® Y by T(x):=N − lim n→∞ f (2 n x) 2 n .Moreover,ifweputm = 0 in (2.11), then we observe that N f (2 n x) 2 n − f (x), n−1 k = 0 t 2 k ≥ N (10δz 0 , t) . Therefore, it follows that N f (2 n x) 2 n − f (x), t ≥ N 10δz 0 , t n−1 k = 0 2 −k ) (2:12) for all x Î X and positive real number t. Next, we show that T is additive. Let x, y Î X and t > 0. Then, we have N T(x + y) − T(x) − T(y), t ≥ min N T(x + y) − f (2 n (x + y)) 2 n , t 4 , N f (2 n x) 2 n − T(x), t 4 , N f (2 n y) 2 n − T(y), t 4 , N f (2 n (x + y)) 2 n − f (2 n x) 2 n − f (2 n y) 2 n , t 4 . (2:13) Since, by (2.8), N f (2 n (x + y)) 2 n − f (2 n x) 2 n − f (2 n y) 2 n , t 4 ≥ N (40δz 0 ,2 n t) , we get lim n→∞ N f (2 n (x + y)) 2 n − f (2 n x) 2 n − f (2 n y) 2 n , t 4 =1 . By the definition of T, the first three terms on the right hand side of the inequality (2.13) tend to 1 as n ® ∞. Therefore, by tending n ® ∞ in (2.13), we observe that T is additive. Najati et al. Journal of Inequalities and Applications 2011, 2011:78 http://www.journalofinequalitiesandapplications.com/content/2011/1/78 Page 5 of 8 Next, we approximate the difference between f and T in a fuzzy sense. For all x Î X and t > 0, we have N( T(x) − f (x), t) ≥ min N T(x) − f (2 n x) 2 n , t 2 , N f (2 n x) 2 n − f (x), t 2 . Since T(x):=N − lim n→∞ f (2 n x) 2 n , letting n ® ∞ in the above inequality and using (N) and (2.12), we get (2.2). It follows from the additivity of T and (2.7) that N( T(x) − g(x)+g(0), t) ≥ min N 2T x 2 − 2f x 2 , t 3 , N 2f x 2 − g(x) − h(0), t 3 , N g(0) + h(0), t 3 ≥ N ( 30δz 0 , t ) for all x Î X and t > 0. So, we get (2.3). Similarly, we can obtain (2.4). To prove the uniqueness of T,letS : X® Y be another additive mapping satisfying the required inequalities. Then, for any x Î X and t > 0, we have N( T(x) − S(x), t) ≥ min N T(x) − f (x), t 2 , N f (x) − S(x), t 2 ≥ N ( 80δz 0 , t ) . Therefore, by the additivity of T and S, it follows that N ( T ( x ) − S ( x ) , t ) = N ( T ( nx ) − S ( nx ) , nt ) ≥ N ( 80δz 0 , nt ) for all x Î X, t >0andn ≥ 1. Hence, the right hand side of the above inequality tends to 1 as n ® ∞. Therefore, T(x)=S(x)forallx Î X. This completes the proof. □ The following is a local Hyers-Ulam stability of the Pexiderized Cauchy functional equation in fuzzy normed spaces. Theorem 2.2. Let X be a normed space,(Y, N) be a fuzzy Banach space, and f, g, h : X® Ybemappingswithf(0) = 0. Suppose that δ >0is a positive real number, and z 0 is a fixed vector of a fuzzy normed space (Z, N’) such that N ( f ( x + y ) − g ( x ) − h ( y ) , t + s ) ≥ min{N ( δz 0 , t ) , N ( δz 0 , s )} (2:14) for all x, y Î X with ||x|| + ||y|| ≥ d and positive real numbers t, s. Then, there exists a unique additive mapping T : X® Y such that N( f (x) − T(x), t) ≥ N (80δz 0 , t) , N( T(x) − g(x)+g(0), t) ≥ N (60δz 0 , t) , N ( T ( x ) − h ( x ) + h ( 0 ) , t ) ≥ N ( 60δz 0 , t ) for all x Î X and t >0. Proof. For the case || x|| + ||y|| <d,letz be an element of X which is defined in the proof of Theorem 2.1. It follows from (N 4 ), (2.5) and (2.14) that Najati et al. Journal of Inequalities and Applications 2011, 2011:78 http://www.journalofinequalitiesandapplications.com/content/2011/1/78 Page 6 of 8 N( f (x + y) − g(x) − h(y ), t + s) ≥ min N f (x + y) − g(y + z) − h(x − z), t + s 5 , N f (x + z) − g(2z) − h(x − z), t + s 5 , N f (y +2z) − g(2z) − h(y), t + s 5 , N f (y +2z) − g(y + z) − h(z), t + s 5 , N f (x + z) − g(x) − h(z), t + s 5 ≥ min{N ( 5δz 0 , t ) , N ( 5δz 0 , s ) } for all x, y Î X with ||x|| + ||y|| <d and positive real numbers t, s. Hence, we have N f (x + y) − g(x) − h(y), t + s ≥ min{N (5δz 0 , t), N (5δz 0 , s) } (2:15) for all x, y Î X and positive real numbers t, s. Letting x =0(y = 0) in (2.15), we get N( f (y) − g(0) − h(y), t + s) ≥ min{N (5δz 0 , t), N (5δz 0 , s)} , N ( f ( x ) − g ( x ) − h ( 0 ) , t + s ) ≥ min{N ( 5δz 0 , t ) , N ( 5δz 0 , s ) } (2:16) for all x, y Î X and positive real numbers t, s. It follows from (2.15) and (2.16) that N( f (x + y) − f (x) − f(y), t + s) ≥ min N f (x + y) − g(x) − h(y), t + s 4 , N f (x) − g(x) − h(0), t + s 4 , N f (y) − g(0) − h(y), t + s 4 , N( g(0) + h(0), t + s 4 ) ≥ min{N ( 20δz 0 , t ) , N ( 20δz 0 , s ) } for all x, y Î X and positive real numbers t, s. The rest of the proof is similar to the proof of Theorem 2.1, and we omit the details. □ Acknowledgements This work was supported by the Korea Research Foundation (KRF) grant funded by the Korea government (MEST) (no. 2009-0075850). Author details 1 Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, 56199-11367 Ardabil, Iran 2 National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, 463-1 Jeonmin-dong, Yuseong-gu, Daejeon 305-811, Korea 3 Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Korea Authors’ contributions All authors carried out the proof. 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Bag, T, Samanta, SK: Finite dimensional fuzzy normed linear spaces. J Fuzzy Math. 11, 687–705 (2003) doi:10.1186/1029-242X-2011-78 Cite this article as: Najati et al.: Local stability of the Pexiderized Cauchy and Jensen’s equations in fuzzy spaces. Journal of Inequalities and Applications 2011 2011:78. Najati et al. Journal of Inequalities and Applications 2011, 2011:78 http://www.journalofinequalitiesandapplications.com/content/2011/1/78 Page 8 of 8 . a generalization of the Hyers’ theorem for additive mappings, and in 1978, Th.M. Rassias [4] succeeded in extending the result of H yers for linear mappings by allowing the Cauchy difference. >0andn ≥ 1. Hence, the right hand side of the above inequality tends to 1 as n ® ∞. Therefore, T(x)=S(x)forallx Î X. This completes the proof. □ The following is a local Hyers-Ulam stability of. Access Local stability of the Pexiderized Cauchy and Jensen’s equations in fuzzy spaces Abbas Najati 1 , Jung Im Kang 2* and Yeol Je Cho 3 * Correspondence: jikang@nims.re. kr 2 National Institute